Mathematics

Systems of equations II

Matrix expression, Rouché-Frobenius theorem, homogeneous systems, and resolution by inverse matrix, Gauss and Cramer.

A system of equations can be represented in matrix form. E.g.: Let the system of equations be:
{a11x+a12y+a13z=b1a21x+a22y+a23z=b2a31x+a32y+a33z=b3{2x−5y+3z=4x−2y+z=35x+y+7z=11\begin{cases} a_{11}x + a_{12}y + a_{13}z = b_1 \\ a_{21}x + a_{22}y + a_{23}z = b_2 \\ a_{31}x + a_{32}y + a_{33}z = b_3 \end{cases} \qquad \begin{cases} 2x - 5y + 3z = 4 \\ x - 2y + z = 3 \\ 5x + y + 7z = 11 \end{cases}
Its matrix expression will be:
(a11a12a13a21a22a23a31a32a33)⋅(xyz)=(b1b2b3)(2−531−21517)⋅(xyz)=(4311)\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \qquad \begin{pmatrix} 2 & -5 & 3 \\ 1 & -2 & 1 \\ 5 & 1 & 7 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \\ 11 \end{pmatrix}
A⋅X=BA \cdot X = B
The square matrix on the left, A, represents the coefficients of the unknowns. The column matrix X is the matrix of unknowns, and the matrix B on the right represents the constants (independent terms).
To study a system, the following matrices are used:
Coefficient matrix: matrix of the coefficients of the unknowns:
A=(2−531−21517)A = \begin{pmatrix} 2 & -5 & 3 \\ 1 & -2 & 1 \\ 5 & 1 & 7 \end{pmatrix}
Augmented matrix: matrix A augmented with the column of constants:
A∗=(2−5341−21351711)A^* = \begin{pmatrix} 2 & -5 & 3 & 4 \\ 1 & -2 & 1 & 3 \\ 5 & 1 & 7 & 11 \end{pmatrix}
A system of equations is said to be:
  • Inconsistent system when it has no solution.
  • Consistent independent system when it has a single, unique solution.
  • Consistent dependent system when it has infinitely many solutions.
Rouché-Capelli Theorem (Also known as Rouché-Frobenius)
Let there be a system of mm equations and nn unknowns. And let AA be its coefficient matrix and A∗A^* its augmented matrix. Then:
  • If rankA=rankA∗=n\text{rank} A = \text{rank} A^* = n, it is a consistent independent system
  • If rankA=rankA∗<n\text{rank} A = \text{rank} A^* < n, it is a consistent dependent system
  • If rankA≠rankA∗\text{rank} A \neq \text{rank} A^*, the system is inconsistent
Systems where the constants are zero. E.g.:
{2x−5y+3z=0x−2y+z=05x+7y+7z=0\begin{cases} 2x - 5y + 3z = 0 \\ x - 2y + z = 0 \\ 5x + 7y + 7z = 0 \end{cases}
It is evident, since rankA=rankA∗\text{rank} A = \text{rank} A^*, that a homogeneous system is always consistent.
It will be independent if rankA=n\text{rank} A = n, with the trivial solution x=0;y=0;z=0x=0; y=0; z=0, and dependent when rankA<n\text{rank} A < n.
Inverse matrix method
If we consider the matrix expression of a system:
A⋅X=B\displaystyle A \cdot X = B
If matrix A is regular, that is, it has an inverse, then the solution of the system is:
X=A−1⋅B\displaystyle X = A^{-1} \cdot B
Gauss method
It consists of transforming the original system into an equivalent one in row echelon form by applying the following equivalent transformations:
  • Multiply (or divide) an equation by a non-zero scalar.
  • Add a linear combination of other equations to an equation.
  • Eliminate an equation that is a linear combination of others in the system.
Example:
{2x−5y+3z=4x−2y+z=35x+y+7z=11A∗=(2−5341−21351711)\begin{cases} 2x - 5y + 3z = 4 \\ x - 2y + z = 3 \\ 5x + y + 7z = 11 \end{cases} \qquad A^* = \begin{pmatrix} 2 & -5 & 3 & 4 \\ 1 & -2 & 1 & 3 \\ 5 & 1 & 7 & 11 \end{pmatrix}
Using elementary transformations it is transformed into:
{x−2y+z=3−y+z=−213z=−26A∗=(1−2130−11−20013−26)\begin{cases} x - 2y + z = 3 \\ -y + z = -2 \\ 13z = -26 \end{cases} \qquad A^* = \begin{pmatrix} 1 & -2 & 1 & 3 \\ 0 & -1 & 1 & -2 \\ 0 & 0 & 13 & -26 \end{pmatrix}
The last line represents the equation 13z=−2613z = -26
The unknowns can be solved from the bottom up.
Cramer's Rule
A linear system of equations is said to be a Cramer system if its coefficient matrix is square and regular, meaning it has a non-zero determinant.
Let there be a system of 2 equations and 2 unknowns, in the following matrix form:
A∗=(a11a12b1a21a22b2)∣A∣=∣a11a12a21a22∣A^* = \begin{pmatrix} a_{11} & a_{12} & b_1 \\ a_{21} & a_{22} & b_2 \end{pmatrix} \qquad |A| = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}
The solution to the system is found by applying:
x=∣b1a12b2a22∣∣A∣y=∣a11b1a21b2∣∣A∣x = \frac{\begin{vmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{vmatrix}}{|A|} \qquad y = \frac{\begin{vmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{vmatrix}}{|A|}
Let there be a system of 3 equations and 3 unknowns, in the following matrix form:
A∗=(a11a12a13b1a21a22a23b2a31a32a33b3)∣A∣=∣a11a12a13a21a22a23a31a32a33∣A^* = \begin{pmatrix} a_{11} & a_{12} & a_{13} & b_1 \\ a_{21} & a_{22} & a_{23} & b_2 \\ a_{31} & a_{32} & a_{33} & b_3 \end{pmatrix} \qquad |A| = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}
The solution to the system is found by applying:
x=∣b1a12a13b2a22a23b3a32a33∣∣A∣y=∣a11b1a13a21b2a23a31b3a33∣∣A∣z=∣a11a12b1a21a22b2a31a32b3∣∣A∣\begin{aligned} x &= \frac{\begin{vmatrix} b_1 & a_{12} & a_{13} \\ b_2 & a_{22} & a_{23} \\ b_3 & a_{32} & a_{33} \end{vmatrix}}{|A|} \\ \\ y &= \frac{\begin{vmatrix} a_{11} & b_1 & a_{13} \\ a_{21} & b_2 & a_{23} \\ a_{31} & b_3 & a_{33} \end{vmatrix}}{|A|} \\ \\ z &= \frac{\begin{vmatrix} a_{11} & a_{12} & b_1 \\ a_{21} & a_{22} & b_2 \\ a_{31} & a_{32} & b_3 \end{vmatrix}}{|A|} \end{aligned}